Improvisation in the Mathematics Classroom

The following is a guest post by Andrea Young, requested by Dr Nic Petty.

Improvisation comedy

Improvisation comedy, or improv for short, is theater that is unscripted.  Performers create characters, stories, and jokes on the spot, much to the delight of audience members.  Surprisingly, the goal of improv is not to be funny!  (Or maybe this isn’t surprising–people trying hard to be funny rarely succeed.)  Rather, improv comedians are encouraged to be “in the moment,” to support their fellow players, and to take risks–the humor follows as a natural consequence.

What does this have to do with mathematics and mathematics education?  If you are a math teacher or professor, you might want to have a classroom where students are deeply engaged with the lesson (i.e. are “in the moment”), actively collaborating with peers (i.e. supporting their fellow players), and willing to make mistakes (i.e. taking risks).  In other words, you want them to develop the skills that improvisers are trained in from their very first improv class.

I started taking improv classes in 2002 at the Hideout Theatre in Austin, TX right around the same time I started a Ph.D. program in mathematics at the University of Texas at Austin.  I realized that the dynamics being developed in my improv classes and troupes were exactly the ones I wanted to develop among the students in my math classes.  So I started using improv games and exercises in my courses.  And I haven’t stopped.  I have now taught mathematics to hundreds of college students, and in every course, I have incorporated some amount of improv.  I have given workshops and presentations to mathematicians, high school teachers, and students about how to use improv to improve group dynamics or to foster communication.   It is powerful to see joy and play cultivated in a college-level mathematics course.  Anecdotally, these techniques work–not for every student, every time–but for enough students enough of the time that I keep using my old favorites and finding new ones to try.

Andrea Young teaches math using Improv principles and games

Some improv exercises to try

Here are three exercises that you might try in your own math classes.  I use these in college classes, but they are easily (and some might argue, more readily) adaptable to younger ages.

Scream circle:  Have the students stand in a circle and put their heads down.  On the count of three, they should all raise their heads and look directly at another student.  If the person they are looking at is also looking at them, both students should scream and leave the circle.  If the other person is not looking at them, they put their head back down.  The game continues until there is only one or two (depending on group size) left.

This exercise is a great way to pair up students to work together.  It also develops the idea of risk-taking because students are encouraged to scream as loud as they can.  It is also quick–depending on the size of the class, this can take fewer than 2 minutes.

Five-headed expert:  Have five students come to the front of the room and stand in a line.  This can be played a few ways.   Here are two:

  1. The students respond to questions one word at a time, as though they are five heads on the same body. Introduce the visiting “expert” and ask them questions, related to course content.  Time permitting, have the class ask questions.
  2. The students respond to questions all in one voice. Otherwise, the game is the same.

This game is a fun way to review concepts and definitions. (For example, what is the limit definition of the derivative?)  It also works on the skills of collaboration and being “in the moment.”  Students must  listen to each other and work together to say things that make sense.

For an example of how this game works in an improv performance, watch this video from the improv group Stranger Things Have Happened.

I am a tree:   Have the students stand in a circle.  One student walks to the center and makes an “I am” statement while striking a pose.  The next student enters the circle and adds to the tableau with another “I am” statement.  A third (and probably final student) enters the tableau like the second.   The professor then clears the tableau, either leaving one of the students to repeat their “I am” statement or not.

This game really highlights the need for collaboration, especially when used in a math context.  I use this as a review or as a way to synthesize concepts. For example, this could be used to review different sets of numbers.  Student one might start with “I am the set of real numbers” and hold his or her arms in a big circle to indicate a set.  Student two could enter the “set” and say, “I am the rationals.”  Another student might intersect the reals with their arms and say, “I am the complex numbers.”

For an introduction to I am a tree, check out this demonstration video from my former improv teacher and troupe mate, Shana Merlin of Merlin Works.

Courage and innovation

I use a lot of active learning techniques in my classes, and I have found improv exercises to be a quick and fun way to develop some of the non-mathematical skills that my students need to be successful in my classroom.  It takes some courage to engage with your students in this way, but I think it is well worth it.

As a final thought, improvisational comedy techniques are not just for students. They can help professional mathematicians become better communicators and more effective teachers. They can even stimulate creativity and problem-solving skills. I encourage you to visit your local comedy theater and to sign up for an improv class.

Andrea Young and fellow trouper performing improvisational musical comedy

Background information and links

Andrea Young is the Special Assistant to the President and Liaison to the Board of Trustees AND an Associate Professor of Mathematical Sciences at Ripon College.  For many years, she performed improv all around the country with Girls, Girls, Girls Improvised Musicals and a variety of other Austin improv troupes.  These days she mostly does community theater, although she regularly improvises silly songs and dances for her toddler.  For more about using improv in math courses, check out

Comment from Dr Nic

Thanks Andrea – it was so great to find someone who was already doing what I was thinking about doing. I would love to hear from other people who have used improv games and techniques in maths and statistics classes. I am learning improv at present, and like the idea of “Yes and…” I will write some more about this in time.


Educating the heart with maths and statistics

What has love got to do with maths?

This morning at the Twitter chat for teachers, (#bfc630nz) the discussion question was, How and what will you teach your students about life this year? As I lurked I was impressed at the ideas and ideals expressed by a mixed bunch of teachers from throughout New Zealand. I tweeted:  “I wonder how often maths teachers think about educating the heart. Yet maths affects how people feel so much.”

My teaching philosophy is summed up as “head, heart and hands”. I find the philosophy of constructivism appealing, that people create their own understanding and knowledge through experiences and reflection. I believe that learning is a social activity, and I am discovering that mathematics is a social endeavour. But underpinning it all I am convinced that people need to feel safe. That is where the heart comes in. “People do not care how much you know until they know how much you care.” Relationships are vital. I wrote previously about the nature of teaching statistics and mathematics.

Teachers are people

In the culture of NZ Maori, when someone begins to address a group of people, they give a mihi, which is an introductory speech following a given structure. The mihi has the role of placing the person with respect to their mountain, their river, their ancestors. It enables the listeners to know who the person is before they begin to speak about anything else. I am not fluent in te reo, so do not give a mihi in Maori (yet), but I do introduce myself so that listeners know who I am. Learners need to know why I am teaching, and how I feel about the subject and about them. It can feel self-indulgent, thinking surely it is about the subject, not about me. But for many learners the teacher is the subject. Just look at subject choices in high school students and that becomes apparent.

Recently I began studying art at an evening class. I am never a passive learner (and for that reason do feel sympathy for anyone teaching me). Anytime I have the privilege of being a learner, I find myself stepping back and evaluating my responses and thinking of what the teacher has done to evoke these responses. Last week, in the first lesson, the teacher gave no introduction other than her name, and I felt the loss. Art, like maths, is emotionally embedded, and I would have liked to have developed more of a relationship with my teacher, before exposing my vulnerability in my drawing attempts. She did a fine job of reassuring us that all of our attempts were beautiful, but I still would like to know who she is.

Don’t sweeten the broccoli

I suspect that some people believe that maths is a dry, sterile subject, where things are right or wrong. Many worksheets give that impression, with columns of similar problems in black and white, with similarly black and white answers. Some attempt to sweeten the broccoli by adding cartoon characters and using bright colours, but the task remains devoid of adventure and creativity. Now, as a child, I actually liked worksheets, but that is probably because they were easy for me, and I always got them right. I liked the column of little red ticks, and the 100% at the end. They did not challenge me intellectually, but I did not know any better. For many students such worksheets are offputting at best. Worksheets also give a limited view of the nature of mathematics.

I am currently discovering how narrow my perception of mathematics was. We are currently developing mathematical activities for young learners, and I have been reading books about mathematical discoveries. Mathematics is full of creativity and fun and adventure, opinion, multiple approaches, discussion and joy. The mathematics I loved was a poor two-dimensional faded version of the mathematics I am currently discovering.I fear most primary school teachers (and possibly many secondary school maths teachers) have little idea of the full potential of mathematics.

Some high school maths teachers struggle with the New Zealand school statistics curriculum. It is embedded in real-life data and investigations. It is not about calculating a mean or standard deviation, or some horrible algebraic manipulation of formulae. Statistics is about observing and wondering, about asking questions, collecting data, using graphs and summary statistics to make meaning out of the data and reflecting the results back to the original question before heading off on another question. Communication and critical thinking are vital. There are moral, ethical and political aspects to statistics.

Teaching mathematics and statistics is an act of social justice

I cannot express strongly enough that the teaching of mathematics and statistics is a political act. It is a question of social justice. In my PhD thesis work, I found that social deprivation correlated with opportunities to learn mathematics. My thoughts are that there are families where people struggle with literacy, but mostly parents from all walks of life can help their children with reading. However, there are many parents who have negative experiences around mathematics, who feel unable to engage their children in mathematical discussions, let alone help them with mathematics homework. And sadly they often entrench mathematical fatalism. “I was no good at maths, so it isn’t surprising that you are no good at maths.”

Our students need to know that we love them. When you have a class of 800 first year university students it is clearly not possible to build a personal relationship with each student in 24 contact hours. However the key to the ninety and nine is the one. If we show love and respect in our dealings with individuals in the class, if we treat each person as valued, if we take the time to listen and answer questions, the other students will see who we are. They will know that they can ask and be treated well, and they will know that we care. When we put time into working out good ways to explain things, when we experiment with different ways of teaching and assessing, when we smile and look happy to be there – all these things help students to know who we are, and that we care.

As teachers of mathematics and statistics we have daunting influence over the futures of our students. We need to make sure we are empowering out students, and having them feel safe is a good start.

Mathematics teaching Rockstar – Jo Boaler

Moving around the education sector

My life in education has included being a High School maths teacher, then teaching at university for 20 years. I then made resources and gave professional development workshops for secondary school teachers. It was exciting to see the new statistics curriculum being implemented into the New Zealand schools. And now we are making resources and participating in the primary school sector. It is wonderful to learn from each level of teaching. We would all benefit from more discussion across the levels.

Educational theory and idea-promoters

My father used to say (and the sexism has not escaped me) “Never run after a woman, a bus or an educational theory, as there will be another one along soon.” Education theories have lifespans, and some theories are more useful than others. I am not a fan of “learning styles” and fear they have served many students ill. However, there are some current ideas and idea-promoters in the teaching of mathematics that I find very attractive. I will begin with Jo Boaler, and intend to introduce you over the next few weeks to Dan Meyer, Carol Dweck and the person who wrote “Making it stick.”

Jo Boaler – Click here for official information

My first contact with Jo Boaler was reading “The Elephant in the Classroom.” In this Jo points out how society is complicit in the idea of a “maths brain”. Somehow it is socially acceptable to admit or be almost defensively proud of being “no good at maths”. A major problem with this is that her research suggests that later success in life is connected to attainment in mathematics. In order to address this, Jo explores a less procedural approach to teaching mathematics, including greater communication and collaboration.

Mathematical Mindsets

It is interesting to  see the effect Jo Boaler’s recent book, “Mathematical Mindsets “, is having on colleagues in the teaching profession. The maths advisors based in Canterbury NZ are strong proponents of her idea of “rich tasks”. Here are some tweets about the book:

“I am loving Mathematical Mindsets by @joboaler – seriously – everyone needs to read this”

“Even if you don’t teach maths this book will change how you teach for ever.”

“Hands down the most important thing I have ever read in my life”

What I get from Jo Boaler’s work is that we need to rethink how we teach mathematics. The methods that worked for mathematics teachers are not the methods we need to be using for everyone. The defence “The old ways worked for me” is not defensible in terms of inclusion and equity. I will not even try to boil down her approach in this post, but rather suggest readers visit her website and read the book!

At Statistics Learning Centre we are committed to producing materials that fit with sound pedagogical methods. Our Dragonistics data cards are perfect for use in a number of rich tasks. We are constantly thinking of ways to embed mathematics and statistics tasks into the curriculum of other subjects.

Challenges of implementation

I am aware that many of you readers are not primary or secondary teachers. There are so many barriers to getting mathematics taught in a more exciting, integrated and effective way. Primary teachers are not mathematics specialists, and may well feel less confident in their maths ability. Secondary mathematics teachers may feel constrained by the curriculum and the constant assessment in the last three years of schooling in New Zealand. And tertiary teachers have little incentive to improve their teaching, as it takes time from the more valued work of research.

Though it would be exciting if Jo Boaler’s ideas and methods were espoused in their entirety at all levels of mathematics teaching, I am aware that this is unlikely – as in a probability of zero. However, I believe that all teachers at all levels can all improve, even a little at a time. We at Statistics Learning Centre are committed to this vision. Through our blog, our resources, our games, our videos, our lessons and our professional development we aim to empower all teacher to teach statistics – better! We espouse the theories and teachings explained in Mathematical Mindsets, and hope that you also will learn about them, and endeavour to put them into place, whatever level you teach at.

Do tell us if Jo Boalers work has had an impact on what you do. How can the ideas apply at all levels of teaching? Do teachers need to have a growth mindset about their own ability to improve their teaching?

Here are some quotes to leave you with:

Mathematical Mindsets Quotes

“Many parents have asked me: What is the point of my child explaining their work if they can get the answer right? My answer is always the same: Explaining your work is what, in mathematics, we call reasoning, and reasoning is central to the discipline of mathematics.”
“Numerous research studies (Silver, 1994) have shown that when students are given opportunities to pose mathematics problems, to consider a situation and think of a mathematics question to ask of it—which is the essence of real mathematics—they become more deeply engaged and perform at higher levels.”
“The researchers found that when students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to explore the problems, they became curious, and their brains were primed to learn new methods, so that when teachers taught the methods, students paid greater attention to them and were more motivated to learn them. The researchers published their results with the title “A Time for Telling,” and they argued that the question is not “Should we tell or explain methods?” but “When is the best time do this?”
“five suggestions that can work to open mathematics tasks and increase their potential for learning: Open up the task so that there are multiple methods, pathways, and representations. Include inquiry opportunities. Ask the problem before teaching the method. Add a visual component and ask students how they see the mathematics. Extend the task to make it lower floor and higher ceiling. Ask students to convince and reason; be skeptical.”

All quotes from

Jo Boaler, Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching

Those who can, teach statistics

The phrase I despise more than any in popular use (and believe me there are many contenders) is “Those who can, do, and those who can’t, teach.” I like many of the sayings of George Bernard Shaw, but this one is dismissive, and ignorant and born of jealousy. To me, the ability to teach something is a step higher than being able to do it. The PhD, the highest qualification in academia, is a doctorate. The word “doctor” comes from the Latin word for teacher.

Teaching is a noble profession, on which all other noble professions rest. Teachers are generally motivated by altruism, and often go well beyond the requirements of their job-description to help students. Teachers are derided for their lack of importance, and the easiness of their job. Yet at the same time teachers are expected to undo the ills of society. Everyone “knows” what teachers should do better. Teachers are judged on their output, as if they were the only factor in the mix. Yet how many people really believe their success or failure is due only to the efforts of their teacher?

For some people, teaching comes naturally. But even then, there is the need for pedagogical content knowledge. Teaching is not a generic skill that transfers seamlessly between disciplines. You must be a thinker to be a good teacher. It is not enough to perpetuate the methods you were taught with. Reflection is a necessary part of developing as a teacher. I wrote in an earlier post, “You’re teaching it wrong”, about the process of reflection. Teachers need to know their material, and keep up-to-date with ways of teaching it. They need to be aware of ways that students will have difficulties. Teachers, by sharing ideas and research, can be part of a communal endeavour to increase both content knowledge and pedagogical content knowledge.

There is a difference between being an explainer and being a teacher. Sal Khan, maker of the Khan Academy videos, is a very good explainer. Consequently many students who view the videos are happy that elements of maths and physics that they couldn’t do, have been explained in such a way that they can solve homework problems. This is great. Explaining is an important element in teaching. My own videos aim to explain in such a way that students make sense of difficult concepts, though some videos also illustrate procedure.

Teaching is much more than explaining. Teaching includes awakening a desire to learn and providing the experiences that will help a student to learn.  In these days of ever-expanding knowledge, a content-driven approach to learning and teaching will not serve our citizens well in the long run. Students need to be empowered to seek learning, to criticize, to integrate their knowledge with their life experiences. Learning should be a transformative experience. For this to take place, the teachers need to employ a variety of learner-focussed approaches, as well as explaining.

It cracks me up, the way sugary cereals are advertised as “part of a healthy breakfast”. It isn’t exactly lying, but the healthy breakfast would do pretty well without the sugar-filled cereal. Explanations really are part of a good learning experience, but need to be complemented by discussion, participation, practice and critique.  Explanations are like porridge – healthy, but not a complete breakfast on their own.

Why statistics is so hard to teach

“I’m taking statistics in college next year, and I can’t wait!” said nobody ever!

Not many people actually want to study statistics. Fortunately many people have no choice but to study statistics, as they need it. How much nicer it would be to think that people were studying your subject because they wanted to, rather than because it is necessary for psychology/medicine/biology etc.

In New Zealand, with the changed school curriculum that gives greater focus to statistics, there is a possibility that one day students will be excited to study stats. I am impressed at the way so many teachers have embraced the changed curriculum, despite limited resources, and late changes to assessment specifications. In a few years as teachers become more familiar with and start to specialise in statistics, the change will really take hold, and the rest of the world will watch in awe.

In the meantime, though, let us look at why statistics is difficult to teach.

  1. Students generally take statistics out of necessity.
  2. Statistics is a mixture of quantitative and communication skills.
  3. It is not clear which are right and wrong answers.
  4. Statistical terminology is both vague and specific.
  5. It is difficult to get good resources, using real data in meaningful contexts.
  6. One of the basic procedures, hypothesis testing, is counter-intuitive.
  7. Because the teaching of statistics is comparatively recent, there is little developed pedagogical content knowledge. (Though this is growing)
  8. Technology is forever advancing, requiring regular updating of materials and teaching approaches.

On the other hand, statistics is also a fantastic subject to teach.

  1. Statistics is immediately applicable to life.
  2. It links in with interesting and diverse contexts, including subjects students themselves take.
  3. Studying statistics enables class discussion and debate.
  4. Statistics is necessary and does good.
  5. The study of data and chance can change the way people see the world.
  6. Technlogical advances have put the power for real statistical analysis into the hands of students.
  7. Because the teaching of statistics is new, individuals can make a difference in the way statistics is viewed and taught.

I love to teach. These days many of my students are scattered over the world, watching my videos (for free) on YouTube. It warms my heart when they thank me for making something clear, that had been confusing. I realise that my efforts are small compared to what their teacher is doing, but it is great to be a part of it.

How to learn statistics (Part 2)

Some more help (preaching?) for students of statistics

Last week I outlined the first five principles to help people to learn and study statistics.

They focussed on how you need to practise in order to be good at statistics and you should not wait until you understand it completely before you start applying. I sometimes call this suspending disbelief. Next I talked about the importance of context in a statistical investigation, which is one of the ways that statistics is different from pure mathematics. And finally I stressed the importance of technology as a tool, not only for doing the analysis, but for exploring ideas and gaining understanding.

Here are the next five principles (plus 2):

6. Terminology is important and at times inconsistent

There are several issues with regard to statistical terminology, and I have written a post with ideas for teachers on how to teach terminology.

One issue with terminology is that some words that are used in the study of statistics have meanings in everyday life that are not the same. A clear example of this is the word, “significant”. In regular usage this can mean important or relevant, yet in statistics, it means that there is evidence that an effect that shows up in the sample also exists in the population.

Another issue is that statistics is a relatively young science and there are inconsistencies in terminology. We just have to live with that. Depending on the discipline in which the statistical analysis is applied or studied, different terms can mean the same thing, or very close to it.

A third language problem is that mixed in with the ambiguity of results, and judgment calls, there are some things that are definitely wrong. Teachers and examiners can be extremely picky. In this case I would suggest memorising the correct or accepted terminology for confidence intervals and hypothesis tests. For example I am very fussy about the explanation for the R-squared value in regression. Too often I hear that it says how much of the dependent variable is explained by the independent variable. There needs to be the word “variation” inserted in there to make it acceptable. I encourage my students to memorise a format for writing up such things. This does not substitute for understanding, but the language required is precise, so having a specific way to write it is fine.

This problem with terminology can be quite frustrating, but I think it helps to have it out in the open. Think of it as learning a new language, which is often the case in new subject. Use glossaries, to make sure you really do know what a term means.

7. Discussion is important

This is linked with the issue of language and vocabulary. One way to really learn something is to talk about it with someone else and even to try and teach it to someone else. Most teachers realise that the reason they know something pretty well, is because they have had to teach it. If your class does not include group work, set up your own study group. Talk about the principles as well as the analysis and context, and try to use the language of statistics. Working on assignments together is usually fine, so long as you write them up individually, or according to the assessment requirements.

8. Written communication skills are important

Mathematics has often been a subject of choice for students who are not fluent in English. They can perform well because there is little writing involved in a traditional mathematics course. Statistics is a different matter, though, as all students should be writing reports. This can be difficult at the start, but as students learn to follow a structure, it can be made more palatable. A statistics report is not a work of creative writing, and it is okay to use the same sentence structure more than once. Neither is a statistics report a narrative of what you did to get to the results. Generous use of headings makes a statistical report easier to read and to write. A long report is not better than a short report, if all the relevant details are there.

9. Statistics has an ethical and moral aspect

This principle is interesting, as many teachers of statistics come from a mathematical background, and so have not had exposure to the ethical aspects of research themselves. That is no excuse for students to park their ethics at the door of the classroom. I will be pushing for more consideration of ethical aspects of research as part of the curriculum in New Zealand. Students should not be doing experiments on human subjects that involve delicate subjects such as abuse, or bullying. They should not involve alcohol or other harmful substances. They should be aware of the potential to do harm, and make sure that any participants have been given full information and given consent. This can be quite a hurdle, but is part of being an ethical human being. It also helps students to be more aware when giving or withholding consent in medical and other studies.

10. The study of statistics can change the way you view the world

Sometimes when we learn something at school, it stays at school and has no impact on our everyday lives. This should not be the case with the study of statistics. As we learn about uncertainty and variation we start to see this in the world around us. When we learn about sampling and non-sampling errors, we become more critical of opinion polls and other research reported in the media. As we discover the power of statistical analysis and experimentation, we start to see the importance of evidence-based practice in medicine, social interventions and the like.

11. Statistics is an inherently interesting and relevant subject.

And it can be so much fun. There is a real excitement in exploring data, and becoming a detective. If you aren’t having fun, you aren’t doing it right!

12. Resources from Statistics Learning Centre will help you learn.

Of course!

Question questions

Ooooh – new data!

There is nothing like a new set of data, just sitting there in the computer, all ready for me to clean and graph and analyse and extract its secrets. I know I should be methodical in my approach, but sometimes I feel like a kid at Christmas, metaphorically ripping open the presents as I jump from graph to procedure, and back to graph again. I then have to go back and do it properly, documenting my approach and recording results, but that’s okay too. That can reveal a second lot of wonders as I sift and ponder.

This is what we should be enabling our students to do. Students need to catch the excitement of making a REAL graph of REAL data and finding out what it REALLY tells them. I have already blogged about the importance of real data in teaching, so those of you who have recently started following you might like to take a look. I also gave some suggestions on how to get real data.

I once dabbled in qualitative research. My PhD thesis used mixed methodology, which entailed recording interviews, transcribing and coding. It seemed like a fun idea at the  time of my research proposal. Sifting through the interviews for gems of insight, getting to figure out common themes and finding linkages and generalities, seems appealing. And it was effective – I came up with a new idea for measuring educational effectiveness through opportunity to learn. But given the choice I won’t be doing it again. I truly admire qualitative researchers, as it takes so much more work than good old quantitative research. Much of it is just slog, reading and coding the interviews. It is really important and totally valid as far as I can see. It’s just that it’s a little – dare I say it – boring.

But I digress. This post is meant to be about questions. The questions you ask in class, the questions in the textbooks, the questions in on-line exercises and the questions in the tests and exams at the end of the unit of work.

In another previous post I lamented how “Statistics Textbooks suck out all the fun.” I cited the work by George Cobb, reviewing textbooks in 1987.

“Judge a book by its exercises and you cannot go far wrong,”  said George Cobb.

It’s still true. The questions are what matter.

I have developed a course for learners who lack confidence in mathematics. There are on-line lecture videos and notes with audio, there are links to other materials, but where the real learning takes place is in the questions. Statistics is not a spectator sport – you have to get in and do it. Things can look easy when you see someone else do them like Olympic diving and producing Pivot-Charts in Excel, playing the piano and developing a linear programming model. But these skills require practice to become proficient. However there is no point in practising the wrong thing, or practising doing the right thing wrongly. Both these can happen when questions and feedback are not well designed.

Recently I have been immersed in questions. I am developing on-line materials for a textbook, and my own on-line materials for supporting high school students and teachers who are struggling with New Zealand’s innovative and world-leading statistics curriculum. From the textbook I have had to select problems to work through in demonstrations. For my own course, I am devising my own questions. As I do this I have become intimately involved with the NCEA questions, (National Certificate of Educational Achievement) as this is how the students will ultimately be tested. This combination has caused me to think a lot about how questions can help or hinder learning.

Students want to pass

Students want to pass

Like it or not, the main motivating force for most students is to pass the test. For some students passing means getting an A or close to 100%, while for others a scrape through is a cause for celebration. But students want to pass. And generally they want to do this with the minimum amount of work. The student attitude is at odds with teachers, who are wanting to increase the amount of work students do, so that they really understand, and catch the excitement, rather than do the minimum to get by. Questions are the way to get them. The questions the students work on should lead them to both goals simultaneously. The students need to feel that their time and effort is moving them toward their goal of passing. The teachers need to engage this effort and seduce the students into seeing how worthwhile and useful, interesting and exciting the subject is.

Good questions

Good questions will do this. A good question will have context, real data and meaning. Statisticians don’t care about x. Mathematicians do. Asking students to interpret the Excel output of a regression of Y on X is a mathematical question and has no place in a statistics textbook or course. Asking how sales are affected by temperature, or grades are affected by time spent doing homework – these are meaningful examples.

A good question needs to test the thing you are trying to test. If you want to know if the student understands the implications of variability, getting them to calculate the standard deviation by hand is not going to do it. If you want a student to know how to use their calculator to find binomial probabilities, then that is what you should ask. But if you want them to be able to identify times when the binomial distribution is a good model of reality, then the question needs to be relevant to that.

There need to be enough questions. By working through multiple examples students come to understand what is specific to each context, and what is general to all examples. This sounds like “drill”, and I am a firm believer in consistent effort on worthwhile questions.

There has to be good feedback. Students need to be able to find out if they are correct, or “on the right track” as so many of my students ask me. Problem is, if you give them the answers, sometimes they just read them and we are back to the “statistics or operations research as a spectator sport” effect. And sometimes students don’t realise the nuances in what they have written, thinking it looks like the model answer, when really they have missed something vital. Often the teacher has to look after this, which requires a lot of time, though we are exploring ways of using on-line quizzes and exercises to enable more targetted feedback, more promptly.

Whatever the approach, we need to make sure that the questions we ask students to work on are leading them to discover the joy of statistics and operations research as well as passing the course.

The End of OR at UC

Blogs are by their nature, personal. Today’s blog is even more personal as I tell of my life with Operations Research and the demise of OR at UC.

Operations Research is a useful, interesting and challenging subject.

I love Operations Research. It was love at first sight, and though I now teach statistics, it is with an attitude strongly shaped by Operations Research.

At school I loved maths. And I was good at it. I captained a team that won the city “Cantamath” competition two years running in the early 1970s. In high school I had a great maths teacher who let me be an assistant to the others in class when I had finished my work. That cemented my desire to be a high-school maths teacher.

I went to university,  intending to become a maths teacher, but unsure about my backup subjects. I took an Economics course, which had six weeks of Operations Research in it. I was sold. This was the subject I had been born for – practical use of numbers to make things better. So I changed my major to Operations Research, and took enough mathematics to still be able to teach at all levels. The lack of numbers and practical application in maths courses above the introductory level left me cold. I added in computer science – and loved programming. My introduction to Statistics was a total mystery, dominated by probability taught through gambling examples, but I managed to get an A anyway.

Fast-forward a couple of decades. I am now at the end of my official operations research career. For twenty years I have been teaching introductory Operations Research and various levels of applied statistics. I completed a PhD thesis on the allocation of resources for the education of students with vision impairment, which used OR methodology and hierarchical linear modeling, though I also dabbled with Data Envelopment Analysis. I have been innovative in my teaching of OR and statistics and won a university teaching award. My videos to teach Excel, Statistics and linear programming have been well-received internationally. I have been fortunate to have worked with many wonderful academics and thousands of mostly wonderful students. (And always wonderful ancillary staff, but don’t get me started on management!)

What I love about Operations Research is the problem-solving practical nature of the work we do. Through student projects I have helped schedule hospital beds and scientific visits to Antarctica. We have helped local government, chicken factories and large trucking firms. We have made things better. When I go to Operations Research conferences I love to hear stories of how OR is helping in less developed countries, and in disaster relief and in so many ways. OR does good. OR makes things better. OR is lots of fun.

So why is my official time with OR over? On Wednesday the Council of the university at which I have been employed voted to close down the Operations Research programme. The university wants to “concentrate” and OR didn’t make the grade, despite two academics taking voluntary redundancy, and a concerted effort to streamline the programme so that it is financially viable. It is the end of an era. In the ultimate irony, the following day I was helping with community outreach and met a student trying to decide what subjects to take. She wanted something that used maths, but wasn’t engineering, and had a people component to it, and possibly was related to business. I told her I knew the perfect subject for her, but that she would not be able to take it our university. I tried to sell her on Operations Management, but I hope I wasn’t too convincing.

So now I will be leaving the university and focussing on bringing statistics to the masses. Statistics is my third love, after mathematics and operations research. I feel a calling to use the operations research way of thinking to help people to understand and enjoy statistics. And thus was born this blog and my future ventures. Statistics is so often taught in a way that confuses people. It is taught by mathematicians who do not understand that most of their students are not. My desire is to help both the teachers and the students so that people understand statistics better. I have not abandoned OR, and we will also be producing materials to help in teaching that. But it might have to step back-stage for a while.

At this sad time I have been enormously buoyed up by comments on my Youtube channel, CreativeHeuristics. Here is a recent one about “Understanding the P-value”

“Thank you for making statistics easy to follow in an entertaining way! You definitely help students like me who have difficulty following the concepts…your method of teaching works perfect for me b/c it helps me get it! I appreciate how your videos simplify the explanations so that its easy to follow. Thank you!”

Comments like this warm the cockles of my heart. It makes it SO worthwhile.

This time is the start of an adventure. I have loved most of my time as an academic, but never really did enough research as I was seduced by the joy of teaching. I am fortunate to have now the opportunity to start a new career after twenty years, and can see the possibilities as well as the hazards. I am enormously fortunate to have a supportive husband and an enthusiastic business partner.

Watch this space!

(And if you want to help us, please buy our apps –  AtMyPace:Statistics and Rogo.)

Statistics Textbooks suck out all the fun.

Do the textbook writers like the students?

In 1987 George Cobb published a paper evaluating statistics textbooks. I am very grateful for it, as it alerted me to the problems with textbooks, and introduced me to the man himself, whose work I greatly admire. Cobb explains that statistics is an inherently interesting and practical subject, but that many textbooks seem to have missed that, or concealed it from the students.

The discipline of statistics is inherently fascinating, applied and important. So why do so many textbooks make it seem mechanistic and abstract? I have been examining textbooks, and wonder if the writers even like their subject matter, or the students they are supposed to be reaching.

I am particularly interested in textbooks for non-mathematicians. The majority of students of statistics are not mathematicians, and are not planning to take any more statistics than they are required to. These students don’t like mathematics. They feel uneasy about taking the course. They are required to take a statistics course as part of their business, psychology or health sciences major. They aren’t even sure why they need to take the course, and hope to get it over and done with and forget about the experience as soon as possible. A previous post talks about how to help students who are feeling negatively towards the course. A textbook for these students needs to get the tone and content right.


A friendly, but authoritative tone is important. Some go too far and become corny in their chattiness. It’s nice to be friendly, but it can be a bit tiresome and the examples can be too cute. But most are just too dry – and have too many words. And far too many equations and algorithms. They seemed bent on protectionism rather than empowerment.


Even more important is the choice of content, and I find this fascinating. I wonder what course some textbooks are designed for. A telling chapter is regression. Regression is an important statistical technique. But what do we tell them about regression? Here is how I have recently seen it done. Provide an example of real data taken from the web. Introduce the problem, then let them wait until the end to find out where you are going. Give the mathematical way of expressing a line, using greek letters. Derive the least squares method of line fitting. Calculate the line by hand. Interpret the slope and the intercept. Calculate the coefficient of determination by hand. Interpret it. Define the residuals, and calculate them. Calculate the F-statistic and t-statistics. Interpret them. Then finish off the story you started at the beginning of the chapter (not that anyone cares anymore).

Some of you may be wondering what is wrong with that. Good – it means I am not preaching to the choir.

Students need to see the whole picture from the beginning. If you absolutely MUST do the mathematics, put it at the end of the chapter for the keen students, but don’t do the maths in the body of the text and scare the others. Do not assume the readers know how to interpret a line. Most don’t. Start with some examples that explain the context, show the line, and explain and apply the model equation. Next work through one example thoroughly, using computer output. Explain the different values and talk about what applies to the sample, and what helps us to generalize to the population. Then provide some more examples, making sure many of them are not statistically significant, some have negative slopes, and all are solving a problem using a sufficiently large sample of real data. Then give them a template for writing up a regression, explaining the different parts. Finally, if you must, you can give them the mathematics. This may keep the instructors happy so that they will buy your book.

There are differing views on finding the mean for ordinal data.

Another telling bit of content is a textbook’s approach to ordinal data. In my video about types of data two instructors argue over whether it is permissible to calculate the mean for ordinal data. It ends with them calling each other “nit-picking mathematician” and “sloppy social scientist”. My approach is to take the middle ground. It is not ideal mathematically to calculate a mean for ordinal data, but much of the time people do, so it is best to know why it may cause problems and that there is an issue, rather than pretending that it never happens. Look in the textbook. I would be wary of any text that states categorically that you cannot find the mean for ordinal data.

There is also the issue of the purpose of the text, both its place in the course, and in the lives of the students. Textbooks can take different roles in courses, largely as a function of the confidence and competence of the instructor. A novice instructor, unsure of the material is well-advised to stick closely to the textbook. But an experienced and engaged instructor will find the text less and less important and more a peripheral second opinion and source of homework exercises. The internet and Wikipedia have replaced the textbook as the source of background knowledge. We suspect a textbook is used more as an expensive combination of talisman and doorstop by the students.

“Judge a book by its exercises and you cannot go far wrong,”  said George Cobb. All exercises in statistics should have context. There is no place for fitting a line by hand calculation to a set of five points with no context. Leave that to mathematics courses. Statistics is about context, and all examples need to reflect that. The data should be real data, so that an interesting result is authentic, not just something dreamed up by the instructor. The data should occasionally be dirty even! (but not too early in the course, without warning). And there should be enough data. Don’t perpetuate bad habits by using too few data.

Having said all this, I do wonder what the role of textbooks is in the education of the future. On-line materials, which can be frequently updated, and crowd-sourced explanations such as found on Wikipedia and elsewhere can fill the place of a textbook.

Or there is always our app – AtMyPace: statistics, which uses video and interactive lessons to teach some important concepts. We are now working to bring this to the web so all can use it. And then maybe I should write a textbook. 😉

You’re teaching it wrong!

“Every year I teach them this and every year they get it wrong!”. This is a phrase I’ve heard from colleagues and from my own mouth. Then it dawned on me – if the students keep getting it wrong, maybe I’m teaching it wrong!

Example of Linear regression analysis

Here’s an example. In linear regression I found that students often had trouble interpreting the slope. They would get it the wrong way around, or just not get it. Every year it was the same and I repeatedly groaned at incorrect interpretations in their work. Then it struck me that maybe it was my fault. Maybe I needed to think harder about why they were not getting it, and implement some changes.

It can be frustrating when students don't seem to get it.

So I did. Consequently we now have a stronger emphasis earlier in the course on fitting lines and interpreting them correctly within multiple real-life contexts. Then later when we have addressed the concepts of hypothesis teaching in multiple contexts, we introduce regression. Students are now equipped to bring together their learning on line-fitting and on inference and hypothesis testing. And, happy day, they do! As a result of the redesigned syllabus, their final written reports are much improved and I am spared the annoyance of repeatedly grading incorrect statements.

What do I mean by “teaching it wrong”?

There are many ways we can teach something poorly (see I do know about adverbs, but “teaching it wrong” is a more memorable phrase). Some of them are given below, with suggested remedies.

We can assume prior understanding

We can make incorrect assumptions about students’ prior understanding. I assumed students understood the meaning of a slope. They probably should have. But they didn’t, and there is no point in berating students or their previous teachers for their deficiency. It doesn’t help. If students need prior knowledge and don’t have it, then we need to teach it. (And not grudgingly!) We may think we don’t have time to teach the earlier material, but it is pointless pressing on if they are not prepared. A quick pre-test can help us assess when students are ready for the new knowledge.

We can miss what the truly tricky aspects are.

When we really understand something, it can be difficult to remember what was difficult. How many of us can remember learning to change gears in a car, a task that becomes automatic? One of the best ways to work out what is difficult is to be there when the students are learning. At university level it is customary to leave the one-on-one  or small group teaching to graduate assistants. The problem is the professors miss out on understanding what is happening to the students in their class. For this reason I always take at least one tutorial group in any class I teach. Grading papers can also help identify what is causing problems.

We can fail to problem solve – staying in our own rut

Teachers need to reflect and experiment. Teachers are smart people, but sometimes we don’t use our smarts well enough in our teaching. It is not good enough to just keep doing what we have always done even if everybody, including the textbook does it too. It is a source of interest to me how many statistics courses and textbooks still teach the normal approximation of the binomial distribution. Fair enough show that the binomial approaches the normal (if you must), but Excel will solve binomial examples just fine for any parameters I’ve given it. There is no need to approximate.

We can fail to give enough good examples

We can fail to give enough examples for students to generalise – or have the examples create incorrect generalisations. A previous post talks about the need for repetition or practice in the construction of knowledge. Given the opportunity, students will entrench wrong interpretations by finding spurious rules and patterns. For example if all the minimizing Linear Program examples have only greater than constraints, students will form the idea that this is what must happen. Or if all examples testing means of weight loss are paired, students will use the context to judge, often erroneously. Well-thought-out sets of examples and exercises can really help, and give the students a sense of unfolding understanding.

The fun part is when you teach it right

Along with many wrong ways, there are many wonderful, right ways we can teach something well. Our task is to find or create these ways, and when we do, the result for learner and teacher is joyful.